Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-2z^2 - 10z - 8}{-6z^2 - 60z - 144}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-2(z^2 + 5z + 4)} {-6(z^2 + 10z + 24)} $ $ y = \dfrac{2}{6} \cdot \dfrac{z^2 + 5z + 4}{z^2 + 10z + 24} $ Simplify: $ y = \dfrac{1}{3} \cdot \dfrac{z^2 + 5z + 4}{z^2 + 10z + 24}$ Next factor the numerator and denominator. $ y = \dfrac{1}{3} \cdot \dfrac{(z + 4)(z + 1)}{(z + 4)(z + 6)}$ Assuming $z \neq -4$ , we can cancel the $z + 4$ $ y = \dfrac{1}{3} \cdot \dfrac{z + 1}{z + 6}$ Therefore: $ y = \dfrac{ z + 1 }{ 3(z + 6)}$, $z \neq -4$